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My answer is (D). It took me 01:55.

(A) While "mathematics" is singular, "the numerical and algebraic relationships" cannot be considered singular. It is plural, so "which is" and "doesn't" do not agree with in number.

(B) "what the word means" is singular. "do not" should be "does not".

(C) An independent clause is typically required after semicolon. One exception is to provide a list of items. Here, an IC is needed but not provided.

(E) "what most people think about" is singular. "are" should be "is".
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The word "mathematics" has no generally accepted definition: the numerical and algebraic relationships, which is how the word is popularly understood, doesn't have much to do with the issues at the forefront of the field.

A) the numerical and algebraic relationships, which is how the word is popularly understood, doesn't have much to do with -> "which is..." (singular verb) doesn't agree with "the numerical and algebraic relationships" (plural). Incorrect.

B) what the word means for most people, numerical and algebraic relationships, do not have a relationship with -> "..the word means..." (singular) doesn't agree with "do not have a ..." (Plural). Incorrect.

C) the popular definition of the word, numerical and algebraic relationships, both irrelevant to -> After colon, we need an IC but, we have a missing verb here. Incorrect.

D) what is popularly understood of the word—numerical and algebraic relationships—has little bearing on -> We have a perfect IC and it makes sense. Let's keep it.

E) what most people think about—numerical and algebraic relationships—are irrelevant to -> "..most people think about.." (singular) doesn't agree with "...are irrelevant to.." (Plural). Incorrect.

So, I think D. :)
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You can find the official explanation by clicking here.
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You can find the official explanation by clicking here.

What a detailed and enriching explanation this is! Even though the issues at hand in the incorrect options weren't difficult to identify even for a non-initiated student like me, but the way you have analyzed all of them is beyond beautiful.

Thank you Profesor generis :) Please never stop these daily butler exercises..

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